where \(n_{\pi /2}\)  means the number of degenerate edge states at \(E=\pm \pi /2\) .
These relations are readily confirmed by comparing the spectrum and
gap functions presented in Figs. REF (b,d) and
Figs. REF (a,c). Notably, with the increase of hopping amplitude
\(J_{1}\) , we observe a series of gap closing and topological phase
transitions in the square-root model. After each transition, the number
of edge modes \(n_{0}\) , \(n_\pi \)  or \(n_{\pi /2}\)  is found to be increased
by 2 even in the presence of NHSE. Specifically, we find \((n_{\pi /2},n_{0},n_{\pi })=(0,0,0)\) ,
\((2,0,0)\) , \((2,2,2)\) , \((2,4,4)\) , \((4,4,4)\) , \((6,4,4)\) , \((6,6,6)\)  with the
increase of \(J_{1}\)  in Fig. REF (d), meanwhile the
winding numbers are \((\nu _{0},\nu _{\pi })=(0,0)\) , \((1,0)\) , \((1,-1)\) ,
\((1,-2)\) , \((2,-2)\) , \((3,-2)\) , \((3,-3)\)  according to the calculation
reported in Ref. [1]}. This process could continue with
the further increase of \(J_{1}\) . We could thus in principle obtain
arbitrarily many topological edge modes at fractional quasienergies
\(E=\pm \pi /2\)  in our square-root non-Hermitian Floquet system. This
highlights the general advantage of Floquet engineering in generating
unique nonequilibrium states with strong topological signatures.
